L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s − 1.03·5-s − 0.999·8-s + (−0.517 + 0.896i)10-s + 0.267·11-s + (0.896 − 1.55i)13-s + (−0.5 + 0.866i)16-s + (3.41 − 5.91i)17-s + (2.19 + 3.79i)19-s + (0.517 + 0.896i)20-s + (0.133 − 0.232i)22-s − 5.46·23-s − 3.92·25-s + (−0.896 − 1.55i)26-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s − 0.462·5-s − 0.353·8-s + (−0.163 + 0.283i)10-s + 0.0807·11-s + (0.248 − 0.430i)13-s + (−0.125 + 0.216i)16-s + (0.828 − 1.43i)17-s + (0.502 + 0.870i)19-s + (0.115 + 0.200i)20-s + (0.0285 − 0.0494i)22-s − 1.13·23-s − 0.785·25-s + (−0.175 − 0.304i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.234i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 + 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.176409415\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.176409415\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 1.03T + 5T^{2} \) |
| 11 | \( 1 - 0.267T + 11T^{2} \) |
| 13 | \( 1 + (-0.896 + 1.55i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.41 + 5.91i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.19 - 3.79i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 5.46T + 23T^{2} \) |
| 29 | \( 1 + (2 + 3.46i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.34 - 5.79i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.73 + 6.46i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.31 + 7.46i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.133 + 0.232i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.378 - 0.656i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.46 + 9.46i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.637 - 1.10i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.31 - 10.9i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.23 + 10.7i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 9.46T + 71T^{2} \) |
| 73 | \( 1 + (-2.70 + 4.69i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.46 - 7.73i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.29 + 5.70i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (3.53 + 6.12i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (9.07 + 15.7i)T + (-48.5 + 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.523784199903799858639942501207, −7.73106472452346061399375813744, −7.12623884949234855364666883981, −5.85378360640521852647841145036, −5.45826548953075352606750129239, −4.33560031159880272351224622048, −3.63625533970583814828468624824, −2.82617742494612125003750554497, −1.65058224693382158802347277447, −0.34770590512463103051771986481,
1.43589135710035617476275694921, 2.83765577357234685836726861634, 3.86507072476603923433679103097, 4.35356984941006989398612551773, 5.47570514401010230782785587493, 6.12204452840794914544791791802, 6.86056791124238131244796747590, 7.85529017499870938658225706338, 8.110586353511309495048990972457, 9.092607234026790599097536890272